Are there intentional objects? Well, are there carentional objects? Let me explain. Suppose we are house hunting, but reject the following houses because
(A) Number 15 lacks a bathroom
(B) Number 18 lacks a kitchen
(C) Number 6 lacks a garden
What is the common property or quiddity or essence of bathrooms at number 15, kitchens at number 18, gardens at number 6? Why, they are ‘things that are lacked’! So let’s define a Latin term ‘carentionality’ to signify the essence or common property of all objects corresponding to the accusative of the verb ‘lacks’! Let’s translate (A) as ‘Number 15’s lacking has a carentional object’ or ‘Number 18 stands in a carentionality relation to some kitchen’.
We could even restate Bill Vallicella's famous aporetic triad as follows:
(1) Some objects lack the nonexistent
(2) Carentionality is a relation between an object, and the object that is lacked
(3) Every R is such that, if R obtains,then all its relata exist
But of course the whole point of the accusative of ‘lacks’ is precisely that there is no object corresponding to it. Otherwise nothing is lacking! The realist will perhaps object that this house is lacking something (a bathroom, a kitchen, a garden). The nominalist agrees, but disagrees that this implies that the house possesses something, or stands in some relation to something, as a result. The logic of ‘is lacking’ is entirely different from, indeed opposite to, the logic of ‘possesses’.
We could even truly say ‘something is lacking in this house – a bathroom’. But that just proves how careful we must be in analysing ordinary English sentences. For we cannot formalise that sentence as ‘for some x, x is in this house and x is lacking’.
Note also the similarity between ‘is lacking’ and ‘is wanting’ as in ‘a single man with a good fortune is in want of a wife’, which is a truth universally acknowledged.
Further thought: there is a whole class of verbs – which in most cases are the same words as the recognised ‘intentional’ verbs, which take an impersonal, non-animate subject. For example
Number 15 lacks a garden
Our bedroom wants a good clean.
That paper deserves an A grade.
This chair is missing a leg.
In all cases, they have the characteristic feature of the properly ‘intentional’ verbs of there not being an object corresponding to their accusative. I.e. there is no garden at no 15, our bedroom has not had a cleaning, the paper didn’t receive an A grade, the chair doesn’t have a leg and so on. In most or all cases we can replace the accusative noun with ‘something’, e.g. if the house lacks a garden, it lacks something, if the chair is missing a leg, it is missing something, and so on. Why is it that these constructions do not tempt us into ‘metaphysical’ theories about the accusative? I’m assuming not, anyway. We aren’t tempted to say that the chair is missing the non-existent. Or to invent a word like ‘carentional’ (from the Latin ‘to lack’) to describe some relation between the chair and some ‘queer’ entity.